Yes, you could do that in Flash as it has more or less all drawing tools that Photoshop does. Alternatively, if your comfort zone lies with Photoshop, you draw your graphics with it and import them in to Flash using the PSD importer feature. On importing, you can still choose to preserve all layers of your PSD.

Last Q - Where would you recommend I go to try to learn how to do this? I'm concerned that there are so many elements of flash that a flash 101 site might show me a lot of things I don't need to know in order to build an interaction like that.

I've learned enough flash that I built a scrolling slideshow that contained pictures that linked to different web pages.

I recommend you go to Flash and work it out as a timeline animation where you use clicking interactions to advance along the timeline. All of what the example you showed is fairly basic stuff that you are not likely to find tutorials dedicated to.

You do not have to pursue it as a timeline design, but in your case it is probably the better way to start, especially if you don't plan a future of working with Flash. If you do plan to work with Flash as a future, then 101 learning can only help.

I don't know what you mean when you say "layer movements" - you cannot move layers. If that is what you are calling timeline animations, then yes, if you want to animate in the timeline you will need to learn how to.

As far as masking goes, you may or may not need any... only you can determine that at this stage.